↑

Properties

Label 2-201-67.66-c4-0-20
Degree $2$
Conductor $201$
Sign $0.655 + 0.754i$
Analytic cond. $20.7773$
Root an. cond. $4.55821$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5.58i·2-s + 5.19i·3-s − 15.1·4-s + 19.0i·5-s + 29.0·6-s − 39.7i·7-s − 4.74i·8-s − 27·9-s + 106.·10-s + 26.8i·11-s − 78.7i·12-s + 248. i·13-s − 221.·14-s − 98.9·15-s − 268.·16-s + 281.·17-s + ⋯
L(s)  = 1  − 1.39i·2-s + 0.577i·3-s − 0.946·4-s + 0.761i·5-s + 0.805·6-s − 0.810i·7-s − 0.0741i·8-s − 0.333·9-s + 1.06·10-s + 0.221i·11-s − 0.546i·12-s + 1.46i·13-s − 1.13·14-s − 0.439·15-s − 1.05·16-s + 0.974·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 + 0.754i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.655 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.655 + 0.754i$
Analytic conductor: \(20.7773\)
Root analytic conductor: \(4.55821\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :2),\ 0.655 + 0.754i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.920898877\)
\(L(\frac12)\) \(\approx\) \(1.920898877\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 5.19iT \)
67 \( 1 + (-3.38e3 + 2.94e3i)T \)
good2 \( 1 + 5.58iT - 16T^{2} \)
5 \( 1 - 19.0iT - 625T^{2} \)
7 \( 1 + 39.7iT - 2.40e3T^{2} \)
11 \( 1 - 26.8iT - 1.46e4T^{2} \)
13 \( 1 - 248. iT - 2.85e4T^{2} \)
17 \( 1 - 281.T + 8.35e4T^{2} \)
19 \( 1 - 293.T + 1.30e5T^{2} \)
23 \( 1 - 651.T + 2.79e5T^{2} \)
29 \( 1 - 34.6T + 7.07e5T^{2} \)
31 \( 1 + 1.29e3iT - 9.23e5T^{2} \)
37 \( 1 - 2.47e3T + 1.87e6T^{2} \)
41 \( 1 - 380. iT - 2.82e6T^{2} \)
43 \( 1 - 1.34e3iT - 3.41e6T^{2} \)
47 \( 1 + 1.21e3T + 4.87e6T^{2} \)
53 \( 1 - 2.84e3iT - 7.89e6T^{2} \)
59 \( 1 + 5.20e3T + 1.21e7T^{2} \)
61 \( 1 + 112. iT - 1.38e7T^{2} \)
71 \( 1 - 8.42e3T + 2.54e7T^{2} \)
73 \( 1 - 8.33e3T + 2.83e7T^{2} \)
79 \( 1 + 4.89e3iT - 3.89e7T^{2} \)
83 \( 1 + 3.33e3T + 4.74e7T^{2} \)
89 \( 1 + 4.74e3T + 6.27e7T^{2} \)
97 \( 1 + 8.42e3iT - 8.85e7T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.25997698662269556750859512029, −10.94897030991133737630783847684, −9.786062599839434019401961741524, −9.375806112844494539909326759187, −7.57857282915469657952156252376, −6.50997957038108077463837187558, −4.66482493465492834403892126195, −3.69449962943420024318380728843, −2.66145730271801342500748732567, −1.07154554523730078703027990100, 0.905818661724032156598056152495, 2.93334904186465758412699866557, 5.19447902910706793427625336627, 5.53364027163110474777472240600, 6.79739831687382976431406053267, 7.894266299024246571483958356759, 8.469547518363564334336504267201, 9.466156087570529230044106897127, 11.04474304913246806457866327117, 12.27907240492270074450770784943

Graph of the $Z$-function along the critical line